gap> TestDeMeoTransitive(2,30);
[  ]

TransitiveIntermediate:=function(d,r)
    local G,H,a,N,s;
    G:=TransitiveGroup(d,r);
    H:=Stabilizer(G,1);
    N:=IntermediateSubgroups(G,H);
    return N;
end;;

TestDeMeoTransitive:=function(d1,d2)
    local LL,d,n,r,int,sub,inc,L,l,i,j,G;
    LL:=[];
    for d in [d1..d2] do
        n:=NrTransitiveGroups(d);
        for r in [1..n] do
            G:=TransitiveGroup(d,r);
            if IsPrimitive(G) and not IsSolvable(G) then
                int:=TransitiveIntermediate(d,r);
                sub:=int.subgroups; inc:=int.inclusions;
                if Length(sub)=5 and Length(inc)=9 then
                   L:=List(Filtered(inc,i->i[1]=0),j->j[2]);
                   if Length(L) = 3 and Length(Filtered(inc,(i->i[1] in L) and i[2]=6)) = 1 then
                       Add(LL,[d,r]);
                   fi;
                fi;
            fi;
        od;
    od;
    return LL;
end;;